Positive solutions for boundary value problem of nonlinear fractional differential equation

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Abstract

In this paper, we investigate the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem: D0+αu(t)+f(t,u(t))=0,0<t<1, u(0)=u(1)=0, where 1<α2 is a real number, D0+α is the standard Riemann–Liouville differentiation, and f:[0,1]×[0,)[0,) is continuous. By means of some fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained. The proofs are based upon the reduction of problem considered to the equivalent Fredholm integral equation of second kind.

Keywords

Fractional differential equation
Boundary value problem
Positive solution
Green's function
Fixed-point theorem

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This work is sponsored by the National Nature Science Foundation of China (10371006) and the Doctoral Program Foundation of Education Ministry of China (1999000722).

1

Present address: College of Information Science and Technology, Shandong University of Science and Technology, Qingdao 266510, People's Republic of China.